Geometry for Elementary School/The Side-Angle-Side congruence theorem

Geometry for Elementary School
The Side-Side-Side congruence theorem	The Side-Angle-Side congruence theorem	The Angle-Side-Angle congruence theorem

The corresponding material in Euclid's elements can be found on page 33 of Book I, Proposition 4 in Issac Todhunter's 1872 translation, The Elements of Euclid for the Use of Schools and Colleges.

In this chapter, we will discuss another congruence theorem, this time the Side-Angle-Side theorem. The angle is called the included angle.

The Side-Angle-Side congruence theorem edit

Given two triangles $\triangle ABC$ and $\triangle DEF$ such that their sides are equal, hence:

The side ${\overline {AB}}$ equals ${\overline {DE}}$ .
The side ${\overline {CA}}$ equals ${\overline {DF}}$ .
The angle $\angle CAB$ equals $\angle FDE$ (These are the angles between the sides).

Then the triangles are congruent and their other angles and sides are equal too. Success!

Proof edit

We will use the method of superposition – we will move one triangle to the other one and we will show that they coincide. We won’t use the construction we learnt to copy a line or a segment but we will move the triangle as whole.

${\text{Superpose }}\triangle ABC{\text{ on }}\triangle DEF{\text{ such that }}A{\text{ is placed on }}D{\text{ and }}{\overline {AB}}{\text{ is placed on }}{\overline {DE}}$
$\because {\overline {AB}}={\overline {DE}}{\text{ (given)}}$
$\therefore B{\text{ coincides with }}E$
$\because \angle CAB=\angle FDE{\text{ (given)}}$
$\therefore {\overline {CA}}={\overline {FD}}$
$\because {\overline {CA}}={\overline {DF}}{\text{ (given)}}$
$\therefore C{\text{ coincides with }}F$
$\therefore {\overline {CB}}={\overline {EF}}$
$\therefore \triangle ABC=\triangle DEF$
$\therefore \triangle ABC\cong \triangle DEF$